tokfandomcom-20200215-history
Pauli equation
In , the Pauli equation or Schrödinger–Pauli equation is the formulation of the for particles, which takes into account the interaction of the particle's with an external . It is the non- limit of the and can be used where particles are moving at speeds much less than the , so that relativistic effects can be neglected. It was formulated by in 1927. Equation For a particle of mass m and electric charge q , in an described by the \mathbf{A} and the \phi , the Pauli equation reads: where \boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z) are the collected into a vector for convenience, and \mathbf{p} = -i\hbar \nabla is the . : |\psi\rangle = \begin{bmatrix} \psi_+ \\ \psi_- \end{bmatrix} is the two-component , a written in . The \hat{H} = \frac{1}{2m} \left- q \mathbf{A}) \right^2 + q \phi is a 2 × 2 matrix because of the . Substitution into the gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See for details of this classical case. The term for a free particle in the absence of an electromagnetic field is just \frac{\mathbf{p}^2}{2m} where is the , while in the presence of an EM field it involves the \mathbf{P} = p - q\mathbf{A} , where \mathbf{P} is the . The Pauli matrices can be removed from the kinetic energy term using the : : (\boldsymbol{\sigma}\cdot \mathbf{a})(\boldsymbol{\sigma}\cdot \mathbf{b}) = \mathbf{a}\cdot\mathbf{b} + i\boldsymbol{\sigma}\cdot \left(\mathbf{a} \times \mathbf{b}\right) to obtain where \mathbf{B} = \nabla \times \mathbf{A} is the magnetic field. Relationship with the Schrödinger equation and the Dirac equation The Pauli equation is non-relativistic, but it does incorporate spin. As such, it can be thought of as occupying the middle ground between: * The familiar Schrödinger equation (on a complex scalar ), which is non-relativistic and does not predict spin. * The Dirac equation (on a ), which is fully (with respect to ) and predicts spin. Note that because of the properties of the Pauli matrices, if the magnetic vector potential \mathbf{A} vanishes, then the equation reduces to the familiar Schrödinger equation for a particle in a purely electric potential ϕ'', except that it operates on a two-component : : \left( \frac{\mathbf{p}^2}{2m} + q \phi \right) \begin{bmatrix} \psi_+ \\ \psi_- \end{bmatrix} = i \hbar \frac{\partial}{\partial t} \begin{bmatrix} \psi_+ \\ \psi_- \end{bmatrix}. Therefore, we can see that the spin of the particle only affects its motion in the presence of a magnetic field. Relationship with Stern–Gerlach experiment Both spinor components satisfy the Schrödinger equation. For a particle in an externally applied \mathbf{B} field, the Pauli equation reads: where : \mathbb{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} is the 2 \times 2 . The can obtain the spin orientation of atoms with one , e.g. silver atoms which flow through an inhomogeneous magnetic field. Analogously, the term is responsible for the splitting of spectral lines (corresponding to energy levels) in a magnetic field as can be viewed in the . References Category:Advanced mathematics